I want to construct a triangle $\Delta(A,B,C)$ with given side $c$ and heights $h_a, h_b$. To construct the triangle means to use only ruler and compass. How can I solve this?
I started as follows:
$1)$ Given the side $c$ with endpoints $A,B$, we can construct the circle with diameter $\overline{AB}$.
$2)$ We can draw a circle with center $A$ and radius $h_a$ and get a Point $F$ as intersection point. My hope is now that the line segment $\overline{AF}$ will become our height $h_a$.
So the third point $C$ has to lie on the line $BF$ such that the height $h_b$ will be the given one. But how can I achieve this?
Best regards
Yes, we have $\overline{AF}=h_a$.
Now, draw another circle with center $B$ and radius $h_b$. Then, you can get a point $G$ which is the intersection point of this circle with the circle with diameter $\overline{AB}$. Here, we have $\overline{BG}=h_b$.
Then, note that $C$ is the intersection point of the line $AG$ with the line $BF$.
Added : The construction depends on the relation between $h_a^2+h_b^2$ and $c^2$.
If $h_a^2+h_b^2=c^2$, then $F=C$.
If $h_a^2+h_b^2\lt c^2$, then $C$ is the intersection point of the line $AG$ with the line $BF$.
If $h_a^2+h_b^2\gt c^2$, then $C$ is the intersection point of the line $AF$ with the line $BG$.